{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:39:49Z","timestamp":1759333189944},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2010,1,1]],"date-time":"2010-01-01T00:00:00Z","timestamp":1262304000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Methods Appl. Math."],"published-print":{"date-parts":[[2010]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We propose a novel numerical method for fast and accurate evaluation\n        of the exchange part of the Fock operator in the Hartree-Fock equation which\n        is a (nonlocal) integral operator. Usually, this challenging computational\n        problem is solved by analytical evaluation of two-electron integrals using the \u201canalytically\n        separable\u201d Galerkin basis functions, like Gaussians. Instead, we employ the\n        agglomerated \u201cgrey-box\u201d numerical computation of the corresponding six-dimensional\n        integrals in the tensor-structured format which does not require analytical separability\n        of the basis set. The point of our method is a low-rank tensor representation of arising\n        functions and operators on an n\u00d7n\u00d7n Cartesian grid and the implementation of the\n        corresponding multi-linear algebraic operations in the tensor product format. Linear\n        scaling of the tensor operations, including the 3D convolution product, with respect to\n        the one-dimension grid size n enables computations on huge 3D Cartesian grids thus\n        providing the required high accuracy. The presented algorithm for evaluation of the\n        exchange operator and a recent tensor method for the computation of the Coulomb\n        matrix are the main building blocks in the numerical solution of the Hartree-Fock equation\n        by the tensor-structured methods. These methods provide a new tool for algebraic\n        optimization of the Galerkin basis in the case of large molecules.<\/jats:p>","DOI":"10.2478\/cmam-2010-0012","type":"journal-article","created":{"date-parts":[[2013,4,15]],"date-time":"2013-04-15T16:10:16Z","timestamp":1366042216000},"page":"204-218","source":"Crossref","is-referenced-by-count":11,"title":["Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods"],"prefix":"10.2478","volume":"10","author":[{"given":"V.","family":"Khoromskaia","sequence":"first","affiliation":[{"name":"1Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany."}]}],"member":"374","container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/10\/2\/article-p204.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.2478\/cmam-2010-0012\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T14:21:39Z","timestamp":1619101299000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/10\/2\/article-p204.xml"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010]]},"references-count":0,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.2478\/cmam-2010-0012","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2010]]}}}